Gauss-Bonnet coupling constant in classically scale-invariant gravity

We discuss the renormalization of higher-derivative gravity, both without and with matter fields, in terms of two primary coupling constants rather than three. A technique for determining the dependence of the Gauss-Bonnet coupling constant on the remaining couplings is explained, and consistency with the local form of the Gauss-Bonnet relation in four dimensions is demonstrated to all orders in perturbation theory. A similar argument is outlined for the Hirzebruch signature and its coupling. We speculate upon the potential implications of instantons on the associated nonperturbative coupling constants.